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Theorem divsfval 14485
Description: Value of the function in divsval 14480. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ercpbl.r  |-  ( ph  ->  .~  Er  V )
ercpbl.v  |-  ( ph  ->  V  e.  _V )
ercpbl.f  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
Assertion
Ref Expression
divsfval  |-  ( ph  ->  ( F `  A
)  =  [ A ]  .~  )
Distinct variable groups:    x,  .~    x, A    x, V    ph, x
Allowed substitution hint:    F( x)

Proof of Theorem divsfval
StepHypRef Expression
1 ercpbl.v . . . . 5  |-  ( ph  ->  V  e.  _V )
2 ercpbl.r . . . . . 6  |-  ( ph  ->  .~  Er  V )
32ecss 7142 . . . . 5  |-  ( ph  ->  [ A ]  .~  C_  V )
41, 3ssexd 4439 . . . 4  |-  ( ph  ->  [ A ]  .~  e.  _V )
5 eceq1 7137 . . . . 5  |-  ( x  =  A  ->  [ x ]  .~  =  [ A ]  .~  )
6 ercpbl.f . . . . 5  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
75, 6fvmptg 5772 . . . 4  |-  ( ( A  e.  V  /\  [ A ]  .~  e.  _V )  ->  ( F `
 A )  =  [ A ]  .~  )
84, 7sylan2 474 . . 3  |-  ( ( A  e.  V  /\  ph )  ->  ( F `  A )  =  [ A ]  .~  )
98expcom 435 . 2  |-  ( ph  ->  ( A  e.  V  ->  ( F `  A
)  =  [ A ]  .~  ) )
106dmeqi 5041 . . . . . . . 8  |-  dom  F  =  dom  ( x  e.  V  |->  [ x ]  .~  )
112ecss 7142 . . . . . . . . . . 11  |-  ( ph  ->  [ x ]  .~  C_  V )
121, 11ssexd 4439 . . . . . . . . . 10  |-  ( ph  ->  [ x ]  .~  e.  _V )
1312ralrimivw 2800 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  V  [ x ]  .~  e.  _V )
14 dmmptg 5335 . . . . . . . . 9  |-  ( A. x  e.  V  [
x ]  .~  e.  _V  ->  dom  ( x  e.  V  |->  [ x ]  .~  )  =  V )
1513, 14syl 16 . . . . . . . 8  |-  ( ph  ->  dom  ( x  e.  V  |->  [ x ]  .~  )  =  V
)
1610, 15syl5eq 2487 . . . . . . 7  |-  ( ph  ->  dom  F  =  V )
1716eleq2d 2510 . . . . . 6  |-  ( ph  ->  ( A  e.  dom  F  <-> 
A  e.  V ) )
1817notbid 294 . . . . 5  |-  ( ph  ->  ( -.  A  e. 
dom  F  <->  -.  A  e.  V ) )
19 ndmfv 5714 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
2018, 19syl6bir 229 . . . 4  |-  ( ph  ->  ( -.  A  e.  V  ->  ( F `  A )  =  (/) ) )
21 ecdmn0 7143 . . . . . 6  |-  ( A  e.  dom  .~  <->  [ A ]  .~  =/=  (/) )
22 erdm 7111 . . . . . . . . 9  |-  (  .~  Er  V  ->  dom  .~  =  V )
232, 22syl 16 . . . . . . . 8  |-  ( ph  ->  dom  .~  =  V )
2423eleq2d 2510 . . . . . . 7  |-  ( ph  ->  ( A  e.  dom  .~  <->  A  e.  V ) )
2524biimpd 207 . . . . . 6  |-  ( ph  ->  ( A  e.  dom  .~ 
->  A  e.  V
) )
2621, 25syl5bir 218 . . . . 5  |-  ( ph  ->  ( [ A ]  .~  =/=  (/)  ->  A  e.  V ) )
2726necon1bd 2679 . . . 4  |-  ( ph  ->  ( -.  A  e.  V  ->  [ A ]  .~  =  (/) ) )
2820, 27jcad 533 . . 3  |-  ( ph  ->  ( -.  A  e.  V  ->  ( ( F `  A )  =  (/)  /\  [ A ]  .~  =  (/) ) ) )
29 eqtr3 2462 . . 3  |-  ( ( ( F `  A
)  =  (/)  /\  [ A ]  .~  =  (/) )  ->  ( F `  A )  =  [ A ]  .~  )
3028, 29syl6 33 . 2  |-  ( ph  ->  ( -.  A  e.  V  ->  ( F `  A )  =  [ A ]  .~  )
)
319, 30pm2.61d 158 1  |-  ( ph  ->  ( F `  A
)  =  [ A ]  .~  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   _Vcvv 2972   (/)c0 3637    e. cmpt 4350   dom cdm 4840   ` cfv 5418    Er wer 7098   [cec 7099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fv 5426  df-er 7101  df-ec 7103
This theorem is referenced by:  ercpbllem  14486  divsaddvallem  14489  divsgrp2  15673  frgpmhm  16262  frgpup3lem  16274  divsrng2  16712  divsrhm  17319
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